On the legendre coefficients of the moments of the general order derivative of an infinitely differentiable function

1995 ◽  
Vol 56 (1-2) ◽  
pp. 107-122 ◽  
Author(s):  
E. H. Doha ◽  
S. I. El-Soubhy
Keyword(s):  
Differentiable Function ◽  
Order Derivative ◽  
General Order ◽  
Infinitely Differentiable Function

scholarly journals The convolution x-r*xs

1976 ◽  
Vol 17 (1) ◽  
pp. 53-56 ◽  
Author(s):  
B. Fisher
Keyword(s):  
Differentiable Function ◽  
Convolution Of Distributions ◽  
Definition Of ◽  
Image Position ◽  
Infinitely Differentiable Function

In a recent paper [1], Jones extended the definition of the convolution of distributions so that further convolutions could be defined. The convolution w1*w2 of two distributions w1 and w2 was defined as the limit ofithe sequence {wln*w2n}, provided the limit w exists in the sense thatfor all fine functions ф in the terminology of Jones [2], wherew1n(x) = wl(x)τ(x/n), W2n(x) = w2(x)τ(x/n)and τ is an infinitely differentiable function satisfying the following conditions:(i) τ(x) = τ(—x),(ii)0 ≤ τ (x) ≤ l,(iii)τ (x) = l for |x| ≤ ½,(iv) τ (x) = 0 for |x| ≥ 1.

The product of the distributions x−r and δ(r−1)(x)

1972 ◽  
Vol 72 (2) ◽  
pp. 201-204 ◽  
Author(s):  
B. Fisher
Keyword(s):  
Differentiable Function ◽  
Open Interval ◽  
Image Position ◽  
Infinitely Differentiable Function

The product of two distributions f and g on the open interval (a, b), where −∞ ≤ a < b ∞, was defined in (1) as the limit of the sequence {fn·gn} provided this sequence is regular in (a, b), wherefor n = 1, 2, … and ρ is a fixed infinitely differentiable function having the following properties:

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